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Equation Attack

Age 16 to 18 Challenge Level:

When you meet a class of equations like this it would be good to find a general formula involving $a$ and $b$ (like the formula for solutions of quadratic equations).

It often helps towards a general solution to try special cases as suggested in this problem and exact solutions can be found in special cases. More generally however there is no closed formula for solutions and you will need to find approximate solutions by numerical methods.

This problem involves the use of logs, solving a quadratic equation and using a numerical method such as interval halving or the Newton Raphson method. To speed up the process of solving this equation for a particular pair of values of $a$ and $b$ you can use a programmable calculator or a spreadsheet.